Since the mid nineteen hundreds, analysis of X-ray diffraction data of crystals has been used extensively for the determination of molecular structure and properties. While the method is employed almost on a routine basis worldwide, it is often a major challenge to identify the three-dimensional structure that best fits the diffraction data. A key obstacle, in particular, is the identification of the phases of the diffracted rays from measurements of intensities alone. This problem is known as the "phase problem" in crystallography and its solution represents a major obstacle towards advancing the frontiers of macromolecular crystallography and structural biology. The primary goal of this project is the development of a systematic methodology for resolving the phase problem in crystallographic computing. Towards this goal, we plan to: (a) develop novel mathematical models for determining three-dimensional crystal structures from single crystal X-ray diffraction measurements; (b) devise mathematical optimization algorithms to solve the above models in a reliable and efficient way, thus increasing the size of tractable structures; (c) develop and make available to the research community a computational system implementing the above models and algorithms; and (d) apply the developed methodology to determine the three-dimensional structures of proteins and other important biological macromolecules. The project will build on a novel algorithm recently pioneered by the Principal Investigator to solve a model that has been demonstrated by the collaborating team to be capable of unraveling structures of biomolecules. The broader impacts of the project include mentoring of graduate students and postdoctoral trainees, integration of research results in a Bioinformatics course, and wide dissemination through on-line software implementing the results of this project. The proposed work promises to lay the foundations of a new generation of crystallographic computing systems that will reveal structures important in the understanding of life, materials science, and drug design. The long term impact to society could be immense as the project could lead to methodology capable of deciphering the secrets of life and playing a pivotal role in the development of new drugs.